Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces

Meyer, Stefan; Wilke, Mathias

doi:10.3934/eect.2013.2.365

Cited by 24 publications

(39 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Kuznetsov equation is one of the models derived from the Navier-Stokes system, and it is well suited for the plane, cylindrical and spherical waves in a fluid [7]. Most of the works on the Kuznetsov equation (1) are treated in the one space dimension [11] or in a bounded spatial domain of R n [12,13,17]. For the viscous case Kaltenbacher and Lasiecka [13] have considered the Dirichlet boundary valued problem and proved for sufficiently small initial data the global well-posedness for n ≤ 3 .…”

Section: Introductionmentioning

confidence: 99%

Cauchy problem for the Kuznetsov equation

Dekkers¹,

Rozanova-Pierrat²

2019

Discrete &Amp; Continuous Dynamical Systems - A

View full text Add to dashboard Cite

We consider the Cauchy problem for a model of non-linear acoustic, named the Kuznetsov equation, describing a sound propagation in thermo-viscous elastic media. For the viscous case, it is a weakly quasi-linear strongly damped wave equation, for which we prove the global existence in time of regular solutions for sufficiently small initial data, the size of which is specified, and give the corresponding energy estimates. In the inviscid case, we update the known results of John for quasi-linear wave equations, obtaining the well-posedness results for less regular initial data. We obtain, using a priori estimates and a Klainerman inequality, the estimations of the maximal existence time, depending on the space dimension, which are optimal, thanks to the blow-up results of Alinhac. Alinhac's blow-up results are also confirmed by a L 2 -stability estimate, obtained between a regular and a less regular solutions.

show abstract

Section: Introductionmentioning

confidence: 99%

Cauchy problem for the Kuznetsov equation

Dekkers¹,

Rozanova-Pierrat²

2019

Discrete &Amp; Continuous Dynamical Systems - A

View full text Add to dashboard Cite

show abstract

“…In some sense these linearizations can be considered as a composition of a heat problem and another linearized problem for the Westervelt equation. While the linearized Westervelt equation can be handled similar as in [MW11,MW13], the heat equation has to be solved with higher regularity conditions. The paper is organized as follows.…”

Section: Introductionmentioning

confidence: 99%

Optimal regularity and exponential stability for the Blackstock–Crighton equation in L p -spaces with Dirichlet and Neumann boundary conditions

Brunnhuber

Meyer

2016

J. Evol. Equ.

Self Cite

View full text Add to dashboard Cite

Abstract. The Blackstock-Crighton equation models nonlinear acoustic wave propagation in monatomic gases. In the present work, we investigate the associated inhomogeneous Dirichlet and Neumann boundary value problems in a bounded domain and prove long-time well-posedness and exponential stability for sufficiently small data. The solution depends analytically on the data. In the Dirichlet case, the solution decays to zero and the same holds for Neumann conditions if the data have zero mean. We choose an optimal L p -setting, where the regularity of the initial and boundary data is necessary and sufficient for existence, uniqueness and regularity of the solution. The linearized model with homogeneous boundary conditions is represented as an abstract evolution equation for which we show maximal L p -regularity. In order to eliminate inhomogeneous boundary conditions, we establish a general higher regularity result for the heat equation. We conclude that the linearized model induces a topological linear isomorphism and then solves the nonlinear problem by means of the implicit function theorem.

show abstract

“…17, 35 and with those for the Kuznetsov equation from Ref. 18, 36, 37, and noting that in Refs 17,35,18,36,. u is the acoustic pressure, i.e., related to ψ by u = ̺ 0 ψ t , we get…”

mentioning

confidence: 84%

“…Consequently, the operator driving the evolution does not exhibit maximal parabolic regularity 27 and the Implicit function theorem argument from, e.g., Ref. 36 cannot be transferred to the present setting.…”

Section: Introductionmentioning

confidence: 91%

The Jordan–Moore–Gibson–Thompson Equation: Well-posedness with quadratic gradient nonlinearity and singular limit for vanishing relaxation time

Kaltenbacher

Nikolić

2019

Math. Models Methods Appl. Sci.

View full text Add to dashboard Cite

In this paper, we consider the Jordan-Moore-Gibson-Thompson equation, a third order in time wave equation describing the nonlinear propagation of sound that avoids the infinite signal speed paradox of classical second order in time strongly damped models of nonlinear acoustics, such as the Westervelt and the Kuznetsov equation. We show well-posedness in an acoustic velocity potential formulation with and without gradient nonlinearity, corresponding to the Kuznetsov and the Westervelt nonlinearities, respectively. Moreover, we consider the limit as the parameter of the third order time derivative that plays the role of a relaxation time tends to zero, which again leads to the classical Kuznetsov and Westervelt models. To this end, we establish appropriate energy estimates for the linearized equations and employ fixed-point arguments for well-posedness of the nonlinear equations. The theoretical results are illustrated by numerical experiments. ,(1.5) respectively. As has been observed in, e.g., Ref. 16, the use of classical Fourier's lawwhere ϑ, q, and K denote the absolute temperature, heat flux vector, and thermal conductivity, respectively, leads to an infinite signal speed paradox, which appears to be unnatural in wave propagation. Therefore in Ref. 16, several other constitutive relations for the heat flux are considered within the derivation of nonlinear acoustic wave equations. Among these is the Maxwell-Cattaneo law τ q t + q = −K∇ϑ,

show abstract

Global well-posedness and exponential stability for Kuznetsov's equation in $L_p$-spaces

Cited by 24 publications

References 14 publications

Cauchy problem for the Kuznetsov equation

Cauchy problem for the Kuznetsov equation

Optimal regularity and exponential stability for the Blackstock–Crighton equation in L p -spaces with Dirichlet and Neumann boundary conditions

The Jordan–Moore–Gibson–Thompson Equation: Well-posedness with quadratic gradient nonlinearity and singular limit for vanishing relaxation time

Contact Info

Product

Resources

About